Module structure question for multiple \wp-functions over multiple Eisenstein series

Determine whether for all integers r \ge 1 and k_1,\dots,k_r \ge 2, the product G_{k_1,\dots,k_r}(\tau)\,\wp_{2}(z;\tau) lies in the Q-linear space generated by multiple \wp-functions together with multiple Eisenstein series (denoted by the authors as {}), which is equivalent to establishing that this space forms a module over \mathcal{E}.

Background

Motivated by analogies with multitangent functions, the authors ask whether the space spanned by multiple \wp-functions (augmented by multiple Eisenstein series) carries a natural \mathcal{E}-module structure. They indicate that this is equivalent to checking that all products of multiple Eisenstein series with \wp_2 lie in the same space.

Resolving this would clarify the algebraic relationship between multiple \wp-functions and multiple Eisenstein series and potentially enable module-theoretic techniques to analyze reductions and relations within the combined space.